Advanced Studies in Pure Mathematics

$n$-nilpotent obstructions to $\pi_1$ sections of $\mathbb{P}^1 - \{0,1,\infty\}$ and Massey products

Kirsten Wickelgren

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Let $\pi$ be a pro-$\ell$ completion of a free group, and let $\pi = [\pi]_1 \supset [\pi]_2 \supset [\pi]_3 \supset \ldots$ denote the lower central series of $\pi$. Let $G$ be a profinite group acting continuously on $\pi$. First suppose that the action is given by a character. Then the boundary maps $\delta_n: H^1 (G,\pi/[\pi]_n)\to H^2 (G, [\pi]_n/[\pi]_{n+1})$ are Massey products. When the action is more general, we partially compute these boundary maps. Via obstructions of Jordan Ellenberg, this implies that $\pi_1$ sections of $\mathbb{P}^1_k - \{0,1,\infty\}$ satisfy the condition that associated $n^{th}$ order Massey products in Galois cohomology vanish. For the $\pi_1$ sections coming from rational points, these conditions imply that $$\langle x^{-1},\ldots,x^{-1},(1-x)^{-1},x^{-1},\ldots,x^{-1}\rangle = 0$$ where $x$ in $H^1 (\mathrm{Gal}(\overline{k}/k), \mathbb{Z}_\ell(\chi))$ is the image of an element of $k^*$ under the Kummer map. For the $\pi_1$ sections coming from rational tangent vectors at infinity, these conditions imply that $$\langle x^{-1},\ldots,x^{-1},(-x)^{-1},x^{-1},\ldots,x^{-1}\rangle = 0.$$

Article information

Galois–Teichmüller Theory and Arithmetic Geometry, H. Nakamura, F. Pop, L. Schneps and A. Tamagawa, eds. (Tokyo: Mathematical Society of Japan, 2012), 579-600

Received: 31 March 2011
Revised: 3 February 2012
First available in Project Euclid: 24 October 2018

Permanent link to this document euclid.aspm/1540417831

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55S30: Massey products
Secondary: 11S25: Galois cohomology [See also 12Gxx, 16H05] 14H30: Coverings, fundamental group [See also 14E20, 14F35]

Massey products group/Galois cohomology section conjecture lower central series


Wickelgren, Kirsten. $n$-nilpotent obstructions to $\pi_1$ sections of $\mathbb{P}^1 - \{0,1,\infty\}$ and Massey products. Galois–Teichmüller Theory and Arithmetic Geometry, 579--600, Mathematical Society of Japan, Tokyo, Japan, 2012. doi:10.2969/aspm/06310579.

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